Ordinary Permutations

Ordinarily, we use the word permutation to designate a one-to-one mapping of a finite set of integers onto itself.  The eight permutations of the numbers 1, 2, 3 are as follows:

1, 2, 3 or (1)(2)(3) (the identity permutation)

2, 1, 3 or (12)(3)
3, 2, 1 or (13)(2)
1, 3, 2 or (1)(23)
(whose squares yield the identity permutation),

and 
2, 3, 1 or (123)
3, 1, 2 or (132)
(whose cubes yield the identity permutation).

Infinite Permutations

In this paper, I suggest an extension to the meaning of the word permutation to include the entire set of natural numbers: 1, 2, 3,. For simplicity, Ill refer to these as infinite permutations.  

I will refer to L2(I) as the following infinite permutation: 2,1,4,3,6,5, , in which consecutive pairs of natural numbers are in reverse order. Thus L2(I)^2 yields the identity element (I, the set of natural numbers). And L3(I) will be: 3,1,2,6,4,5, , in which consecutive triples of natural numbers are reordered such that the largest is placed before the other two.  Thus L3(I)^3 also yields the identity element.  (NOTE: I could easily define another entire set of these functions which generate right-shifting permutations.  Thus R2(I) = L2(I), but R3(I) = 2,3,1,5,6,4, = L3(I)^2. But the set of left-shifting permutations are easier to work with for my purposes here.)

Generating finite permutations using the Ln(I) functions

For any ordinary permutation (i.e. a finite permutation) we can determine a set of functions Ln(I) which generate it.

For example, lets look at the set of six permutations we mentioned earlier.

1,2,3   is derived from the first three natural numbers in the set I, which we could refer to as L1(I), another term for the identity.

1,3,2   is derived from the first three numbers in the infinite permutation defined by: L3(L4(L2(I))) = 1,3,2,6,4,7,11,5,8,12,10,9,

2,1,3  is derived from the first three numbers in the infinite permutation defined by: L4(L2(L3(I))) = 2,1,3,6,9,5,4,7,10,12,8,11,

2,3,1   is derived from the first three numbers in the infinite permutation defined by: L3(I)^2 = 2,3,1,5,6,4,8,9,7,11,12,10,

3,1,2   is derived from the first three numbers in the infinite permutation defined by: L3(I) = 3,1,2,6,4,5,9,7,8,12,10,11,

3,2,1   is derived from the first three numbers in the infinite permutation defined by: L4(L2(I)) = 3,2,1,4,7,6,5,8,11,10,9,12,

Correspondence between infinite permutations and rational numbers


Finite permutations can be listed in a particular order, which means that they form a countably infinite (or denumerable set). First we list the 2 permutations of length 2, then the 6 permutations of length 3, then the 24 permutations of length 4, and so on, as shown here:


1,2
2,1
1,2,3
1,3,2
2,1,3
2,3,1
3,1,2
3,2,1
1,2,3,4
1,2,4,3
1,3,2,4
1,3,4,2
1,4,2,3
1,4,3,2
2,1,3,4
2,1,4,3
2,3,1,4
2,3,4,1
2,4,1,3
2,4,3,1
3,1,2,4
3,1,4,2
3,2,1,4
3,2,4,1
3,4,1,2
3,4,2,1
4,1,2,3
4,1,3,2
4,2,1,3
4,2,3,1
4,3,1,2
4,3,2,1
Etc.


And as I demonstrated in the previous section, each of these permutations can be associated with an infinite permutation defined by the Ln(I) functions.  

1,2   --> L1(I)
2,1   --> L2(I)
1,2,3 --> L1(I)
1,3,2 --> L3(L4(L2(I)))
2,1,3 --> L4(L2(L3(I)))
2,3,1 --> L3(I)^2
3,1,2 --> L3(I)
3,2,1 --> L4(L2(I))

Each of these infinite permutations has a repeating pattern associated with it which can be associated with a rational number.  All we need to do is replace the natural numbers by their counterparts modulo 10, and join them into a long string of digits following a decimal point.

Then the first few terms in the sequence are:

L1(I) --> 0.12345678901234567890
L2(I) --> 0.21436587092143658709
L3(L4(L2(I))) --> 0.132647158209354869370421576081592643798203714865910425936087
L4(L2(L3(I))) --> 0.213695470281435817692403658039814625870251036847092473258069
L3(I)^2 --> 0.231564897120453786019342675908231564897120453786019342675908
L3(I) --> 
0.312645978201534867190423756089312645978201534867190423756089

L4(L2(I)) -->
0.321476581092543698703214765810925436987032147658109254369870
L4(L4(L5(L3(I)))) -->
0.124359624078391571864270563943086212785915408639707139420856
L4(L5(L3(L4(I)))) -->
0.132425189673804647501693522869731015748083945227960083167459
L4(L5(L3(L2(I)))) -->
0.134205169873026427581893540649731215760863925407982803147659
L4(L6(L3(L3(I)))) -->
0.142385602971364507824193586729046315708941268537920163480759
L4(L4(L4(L2(I)))) -->
0.143258769210365470981432587692103654709814325876921036547098
L5(L3(L4(L4(I)))) -->
0.213480715662935704935824117826659044379028371168599260573480



